defpred S₁[ Nat] means (($1 ^2) - $1) + 1 > 0 ;
A1: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume ((k ^2) - k) + 1 > 0 ; :: thesis: S₁[k + 1]
then (((k ^2) - k) + 1) + (2 * k) > 0 + 0 ;
hence S₁[k + 1] ; :: thesis: verum

end;

A2: S₁[ 0 ] ;
for n being Nat holds S₁[n] from NAT_1:sch 2(A2, A1);
hence for n being Element of NAT holds ((n ^2) - n) + 1 > 0 ; :: thesis: verum